On the existence and convergence of pseudomoments for variables in the domain of normal attraction of an operator stable distribution

Abstract

ABSTRACT. Integrals are constructed to replace absolute moments for variables in the domain of normal attraction of an operator stable law. These integrals, called pseudomoments, improve on the geometric information contained in absolute moments. Existence and convergence to appropriate values of these integrals are shown for the variables and their affine normalized sums. Introduction.The purpose of this note is to prove the existence and convergence of certain integrals (called pseudomoments) involving weakly convergent affine normalized sums of independent, identically distributed random vectors taking values in a finite-dimensional Euclidean space, when the normalizing operators take on a special form. The existence of these pseudomoments for the common distribution of the summands, and the convergence of the pseudomoments of the affine normalized sums to the corresponding pseudomoments of the limiting distribution, are shown to occur precisely when the latter exist, in the case of a non-Gaussian limit. The case of Gaussian limits is also covered.The pseudomoments of a distribution (for definitions, see §2) are designed to reflect the geometry of the distribution more accurately and informatively than do ordinary absolute moments. Since the great utility of affine normalization is its attention to variable decay rates in the probability tails of the one-dimensional projections (marginals) of a distribution, it is clearly an advantage to construct moment-integrals showing similar attention to these marginal tails, as we will see pseudomoments do.

Pseudomoments for Generalized Domains of Attraction

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In this paper we resolve a conjecture of Weiner (1987) concerning the existence of the so-called pseudomoments for a random vector which belongs to the generalized domain of attraction of an operator-stable law. The proof is a direct consequence of the spectral decomposition theorem for generalized domains of attraction in Meerschaert (1991).

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“…The norming operators n may perform rotations as well as scaling. In order to incorporate this geometry into the study of moment behavior for operator-stable laws and generalized domains of attraction, Weiner (1987) introduced the concept of pseudomoments.…”